Integro-differential Equations Of Fractional Order Research Articles

Application of an efficient analytical technique based on Aboodh transformation to solve linear and non-linear dynamical systems of integro-differential equations

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). In the current work, the solutions to linear and non-linear FIDEs and their systems have been analyzed by using the Aboodh transform decomposition method (ATDM). The Aboodh transformation is first utilized to simplify the given problem, and then the decomposition method is implemented to obtain the required results. The Adomian polynomials and Daftardar–Jafari polynomials are used to control the non-linear term in the system. The obtained results are compared with both polynomials and with different fractional orders. We show that Daftardar–Jafari polynomials work more accurately for the proposed method than Adomian polynomials. The graphical and tabular representations are made to show the effectiveness of the proposed technique. The accuracy of ATDM is remarkable while using the proposed technique. It is observed that ATDM solutions very closed to the exact solutions of the problems. ATDM has a lower computation cost and higher rate of convergence and thus can be used to solve other non-linear fractional order integro-differential equations and their systems. Finally, we propose some future research directions by utilizing ATDM.

Numerical Solution of Fractional Order Integro-Differential Equations via Müntz Orthogonal Functions

In this paper, we derive a spectral collocation method for solving fractional-order integro-differential equations by using a kind of Müntz orthogonal functions that are defined on 0,1 and have simple and real roots in this interval. To this end, we first construct the operator of Riemann–Liouville fractional integral corresponding to this kind of Müntz functions. Then, using the Gauss–Legendre quadrature rule and by employing the roots of Müntz functions as the collocation points, we arrive at a system of algebraic equations. By solving this system, an approximate solution for the fractional-order integro-differential equation is obtained. We also construct an upper bound for the truncation error of Müntz orthogonal functions, and we analyze the error of the proposed collocation method. Numerical examples are included to demonstrate the validity and accuracy of the method.

Spectrum of an Integro-Differential Equation of Fractional Order

Spectrum of an Integro-Differential Equation of Fractional Order

Periodic Solutions for an Impulsive System of Fractional Order Integro-Differential Equations with Maxima

Periodic Solutions for an Impulsive System of Fractional Order Integro-Differential Equations with Maxima

A hybrid technique for approximating the solution of fractional order integro differential equations

In this article, we present an effective approach for solving nonlinear fractional order integro-differential equations. The fractional order derivative will be in the Caputo sense. For this, we propose an approach combined the least squares method together with the Laplace transform and the shifted Legendre polynomials. Using the proposed approach we will converts the nonlinear fractional order integro-differential equation into a se of (N+1) algebraic equations, where the solution to the resultant equation provides us with the unknown coefficients of the infinite series that have been used to approximate the solution to the considered fractional order integro-differential equations. From the results of the given examples, one can conclude that the recommended approach in this article is simple to implement and accurate.

A developed technique of homotopy analysis method for solving nonlinear systems of Volterra integro-differential equations of fractional order

In this paper, a developed technique of homotopy analysis method (HAM) is presented to solve nonlinear system of fractional order Volterra integro-differential equations. The fractional derivative is described in the Caputo form. The main thing we have done in this paper is that we have simplified the process of evaluating the fractional order integrations in the deformation equations than the standard HAM. The proposed technique is tested by some illustrative examples in order to demonstrate its accuracy, applicability and efficiency.

Adaptation of Caputo residual power series scheme in solving nonlinear time fractional Schrödinger equations

This study regards the time-fractional nonlinear Schrödinger equations (TFNLSEs). The TFNLSE is used to depict nonlocal quantum phenomena in quantum physics and investigate the quantum behaviors of either long-range relations or time-dependent processes with many scales. Basic definitions of fractional derivatives are described in the Caputo fractional derivative (CaFD) sense. We utilize the residual power series method (RPSM) to achieve the approximate solutions of the TFNLSE. The RPSM is a numeric analytic technique for solving various ordinary, partial, and fuzzy differential and integrodifferential equations of fractional order. It is a practical optimization approach, supplying solutions in the closed form of known functions. The primary benefit of this technique is the clarity in calculating the coefficients of terms of the series solution by employing only the differential operators and not the other well-known analytic approaches that require integration operators, which are challenging in the fractional case. Our main aim in this paper is to find approximate solutions using RPSM and compare these approximate solutions with exact solutions. We discuss three applications, showing the validity, precision, and efficiency of the RPSM. The outcomes are presented in graphs and tables, which display the method’s effectiveness, quality, and strength. The RPSM is entirely compatible with the complexity of this problem, and the accepted outcomes are highly favorable. The offered technique can be easily used in the spaces of higher-dimension solutions without limiting the nature of the equation or the category style. Also, this method can be implemented as a novel alternative technique to deal with many nonlinear issues occurring in quantum physics.

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

Extensions of some differential inequalities for fractional integro-differential equations via upper and lower solutions

This paper deals with some differential inequalities for generalized fractional integro-differential equations by using the technique of upper and lower solutions. The fractional differential operator is taken in Caputo’s sense and the nonlinear term divided into two parts depends on the fractional integrals of an unknown function with two different fractional orders. The results are studied by employing a variety of coupled upper and lower solutions. These theorems have some potential for extending the iterative techniques to fractional order integro-differential equations and to coupled systems of integro-differential fractional equations to obtain the existence of solutions as well as approximate solutions for the considered problem.

Study on Urban Proximity Prediction Based on Doppler Radar Gust Front Characteristics and Urban Microclimate Characteristics

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Controllability and Observability Criteria for Systems Described by Fractional Differential Equations

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Existence and data dependence results for neutral fractional order integro-differential equations

<abstract><p>We assess the multi-derivative nonlinear neutral fractional order integro-differential equations with Atangana-Baleanu fractional derivative of the Riemann-Liouville sense. We discuss results about the existence and difference solution on some data, based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ with generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and the Gronwall-Bellman inequality.</p></abstract>

Second kind Chebyshev collocation technique for Volterra-Fredholm fractional order integro-differential equations

In this work, we present the numerical solution of fractional order Volterra–Fredholm integro-differential equations using the second kind of Chebyshev collocation technique. First, we transformed the problem into a system of linear algebraic equations, which are then solved using matrix inversion to obtain the unknown constants. Furthermore, numerical examples are used to outline the method’s accuracy and efficiency using tables and figures. The results show that the method performed better in terms of improving accuracy and requiring less rigorous work.©2022 JNSMR UIN Walisongo. All rights reserved.

Numerical solution of systems of fractional order integro-differential equations with a Tau method based on monic Laguerre polynomials

In this paper, numerical technique based on monic Laguerre polynomials is proposed to obtain approximate solutions of initial value problems for systems of fractional order integro-differential equations (FIDEs). Operational fractional integral matrix is constructed. This operational matrix is applied together with the monic Laguerre Tau method to solve systems of FIDEs. This systems of FIDEs will be transformed into a system of algebraic equations which can be solved easily. Numerical results and comparisons with other methods are also presented to show the efficiency and applicability of the proposed method.

A modified reproducing Kernel Hilbert space method for solving fuzzy fractional integro-differential equations

The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] \bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method

The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis

The Bernoulli polynomials reproducing kernel method for nonlinear Volterra integro-differential equations of fractional order with convergence analysis

Solving Fredholm Integro-Differential Equation of Fractional Order by Using Sawi Homotopy Perturbation Method

In our article, we develop an efficient compined method consisting of integral transform and Homotopy Perturbation Method for solving Fractional Integro-differential equations of Fredholm type. Sawi transformation formulas of fractional types are derived and used in the proposed method. The Caputo derivative is the fractional derivative type that we used. The results of numerical experiments show the efficiency of our newly developed method. Exact solution and the existing method Homotopy perturbation method is used in the comparison with our new results.

Numerical solution of fractional-order integro-differential equations using Laguerre wavelet method

Laguerre wavelet based numerical method is used for the solution of fractional order Integro-differential equations. This method is based on Laguerre wavelet basis. Laguerre wavelet method (LWM) is then utilized to reduce the fractional Integro-differential equations into the system of algebraic equations. Illustrative examples are represented with tables and figures to check the efficiency and applicability of the proposed technique.

Modification of Sumudu Decomposition Method for Nonlinear Fractional Volterra Integro-Differential Equations

In this paper, modification of Sumudu decomposition method is successfully applied to find the approximate solution of nonlinear Volterra integro-differential equation of fractional order. The proposed method is based on the combining of two powerful techniques, Sumudu decomposition method and Daftardar-Gejji and Jafari (DGJ) method. Several illustrative examples are given to demonstrate the validity, reliability, and efficiency of the proposed technique.

Numerical solution of system of fuzzy fractional order Volterra integro-differential equation using optimal homotopy asymptotic method

<abstract> <p>In this paper, an efficient technique called Optimal Homotopy Asymptotic Method has been extended for the first time to the solution of the system of fuzzy integro-differential equations of fractional order. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The fuzzy fractional derivatives are defined in Caputo sense. It is followed by suggesting a new result from Optimal Homotopy Asymptotic Method for Caputo fuzzy fractional derivative. We then construct a detailed procedure on finding the solutions of system of fuzzy integro-differential equations of fractional order and finally, we demonstrate a numerical example. The validity and efficiency of the proposed technique are demonstrated via these numerical examples which depend upon the parametric form of the fuzzy number. The optimum values of convergence control parameters are calculated using the well-known method of least squares, obtained results are compared with fractional residual power series method. It is observed from the results that the suggested method is accurate, straightforward and convenient for solving system of fuzzy Volterra integrodifferential equations of fractional order.</p> </abstract>